Let $\mathrm{A}=\{1,2,3,4,5,6\}$. The number of one-one functions $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{A}$ such that $\mathrm{f}(1) \geq 3, \mathrm{f}(3) \leq 4$ and $f(2)+f(3)=5$, is ______.

JEE Main · 2026 · 05 Apr 2026 · Shift 1 · Sets Relations and Functions

Let

A = {1, 2, 3, 4, 5, 6}

. The number of one-one functions

f : A \to A

such that

f (1) \geq 3, f (3) \leq 4

and

f (2) + f (3) = 5

, is ______.

Answer: 72

The condition

f (2) + f (3) = 5

with distinct images and

f (3) \leq 4

allows the pairs

(f (2), f (3)) = (4, 1), (1, 4), (3, 2), (2, 3)

. In each pair two values are used;

f (1)

must satisfy

f (1) \geq 3

while avoiding those two values, which leaves exactly

3

admissible choices every time. The three remaining elements are then assigned to

f (4), f (5), f (6)

3! = 6

ways. Each pair therefore yields

3 \times 6 = 18

functions, and

4 \times 18 = 72

. The number of such one-one functions is

72

JEE Main · 2026 · 05 Apr 2026 · Shift 1 · Integer answer